In this paper we give a new and elementary proof to the following fact: each closed orientable surface of positive genus admits a both chaotic and expansive homeomorphism. Further more, we show that the homeomorphisms given are also weakly mixing.

This lecture note is mainly about arithmetic progressions, different regularity lemmas and removal lemmas. We will be very brief most of the time, trying to avoid technical details, even definitions. For most technical details, we refer the reader to references. Apart from arithmetic progressions, we also discuss property testing and extremal graph theory.

Starting from the free field realization of Kac–Moody Lie algebra, we define a generalized Yang–Yang function. Then for the Lie algebra of type $A_{n}$, we derive braiding and fusion matrix by braiding the thimble from the generalized Yang–Yang function. One can construct a knots invariant H(K) from the braiding and fusion matrix. It is an isotropy invariant and obeys a skein relation. From them, we show that the corresponding knots invariant is HOMFLY polynomial.

Analysis-suitable T-splines are a topological-restricted subset of T-splines, which are optimized to meet the needs both for design and analysis (Li and Scott Models Methods Appl Sci 24:1141–1164, 2014; Li et al. Comput Aided Geom Design 29:63–76, 2012; Scott et al. Comput Methods Appl Mech Eng 213–216, 2012). The paper independently derives a class of bi-degree $(d_{1}, d_{2})$ T-splines for which no perpendicular T-junction extensions intersect, and provides a new proof for the linearly independence of the blending functions. We also prove that the sum of the basis functions is one for an analysis-suitable T-spline if the T-mesh is admissible based on a recursive relation.

We study optimal insider control problems, i.e., optimal control problems of stochastic systems where the controller at any time t, in addition to knowledge about the history of the system up to this time, also has additional information related to a future value of the system. Since this puts the associated controlled systems outside the context of semimartingales, we apply anticipative white noise analysis, including forward integration and Hida–Malliavin calculus to study the problem. Combining this with Donsker delta functionals, we transform the insider control problem into a classical (but parametrised) adapted control system, albeit with a non-classical performance functional. We establish a sufficient and a necessary maximum principle for such systems. Then we apply the results to obtain explicit solutions for some optimal insider portfolio problems in financial markets described by Itô–Lévy processes. Finally, in the Appendix, we give a brief survey of the concepts and results we need from the theory of white noise, forward integrals and Hida–Malliavin calculus.

We give a differential-geometric construction of Calabi–Yau fourfolds by the ‘doubling’ method, which was introduced in Doi and Yotsutani (N Y J Math 20:1203–1235, 2014) to construct Calabi–Yau threefolds. We also give examples of Calabi–Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are admissible pairs, which were first dealt with by Kovalev (J Reine Angew Math 565:125–160, 2003). Here in this paper an admissible pair $(\overline{X},D)$ consists of a compact Kähler manifold $\overline{X}$ and a smooth anticanonical divisor D on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ with $\dim _{\mathbb {C}}\overline{X}_i=4$ satisfy the gluing condition, we can glue $\overline{X}_1\setminus D_1$ and $\overline{X}_2\setminus D_2$ together to obtain a compact Riemannian 8-manifold (M, g) whose holonomy group $\mathrm {Hol}(g)$ is contained in $\mathrm {Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of M equals 2, then M is a Calabi–Yau fourfold, i.e., a compact Ricci-flat Kähler fourfold with holonomy $\mathrm {SU}(4)$. In particular, if $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ are identical to an admissible pair $(\overline{X},D)$, then the gluing condition holds automatically, so that we obtain a compact Riemannian 8-manifold M with holonomy contained in $\mathrm {Spin}(7)$. Moreover, we show that if the admissible pair is obtained from any of the toric Fano fourfolds, then the resulting manifold M is a Calabi–Yau fourfold by computing $\widehat{A}(M)=2$.